chapter 7 the cartan–dieudonn´e theoremcis610/cis61009sl5.pdf256 chapter 7. the...

Chapter 7

The Cartan–Dieudonne Theorem

7.1 Orthogonal Reflections

Orthogonal symmetries are a very important example ofisometries. First let us review the definition of a (linear)projection.

Given a vector space E, let F and G be subspaces of Ethat form a direct sum E = F ⊕G.

Since every u ∈ E can be written uniquely asu = v + w, where v ∈ F and w ∈ G, we can define thetwo projections pF :E → F and pG:E → G, such that

pF (u) = v and pG(u) = w.

255

256 CHAPTER 7. THE CARTAN–DIEUDONNE THEOREM

It is immediately verified that pG and pF are linear maps,and that p2

F = pF , p2G = pG, pF ◦ pG = pG ◦ pF = 0, and

pF + pG = id.

Definition 7.1.1 Given a vector space E, for any twosubspaces F and G that form a direct sum E = F ⊕G,the symmetry with respect to F and parallel to G, orreflection about F is the linear map s:E → E, definedsuch that

s(u) = 2pF (u) − u,

for every u ∈ E.

Because pF + pG = id, note that we also have

s(u) = pF (u) − pG(u)

ands(u) = u− 2pG(u),

s2 = id, s is the identity on F , and s = −id on G.

7.1. ORTHOGONAL REFLECTIONS 257

We now assume that E is a Euclidean space of finitedimension.

Definition 7.1.2 Let E be a Euclidean space of finitedimension n. For any two subspaces F and G, if F andG form a direct sum E = F ⊕ G and F and G areorthogonal, i.e. F = G⊥, the orthogonal symmetry withrespect to F and parallel to G, or orthogonal reflectionabout F is the linear map s:E → E, defined such that

s(u) = 2pF (u) − u,

for every u ∈ E.

When F is a hyperplane, we call s an hyperplane symme-try with respect to F or reflection about F , and whenG is a plane, we call s a flip about F .

It is easy to show that s is an isometry.


u

s(u)

pG(u)

−pG(u)

pF (u)

F

G

Figure 7.1: A reflection about a hyperplane F

Using lemma 5.2.7, it is possible to find an orthonormalbasis (e1, . . . , en) of E consisting of an orthonormal basisof F and an orthonormal basis of G.

Assume that F has dimension p, so that G has dimensionn− p.

With respect to the orthonormal basis (e1, . . . , en), thesymmetry s has a matrix of the form

(Ip 00 −In−p

)


Thus, det(s) = (−1)n−p, and s is a rotation iff n − p iseven.

In particular, when F is a hyperplane H , we havep = n − 1, and n − p = 1, so that s is an improperorthogonal transformation.

When F = {0}, we have s = −id, which is called thesymmetry with respect to the origin. The symmetrywith respect to the origin is a rotation iff n is even, andan improper orthogonal transformation iff n is odd.

When n is odd, we observe that every improper orthogo-nal transformation is the composition of a rotation withthe symmetry with respect to the origin.


When G is a plane, p = n− 2, and det(s) = (−1)2 = 1,so that a flip about F is a rotation.

In particular, when n = 3, F is a line, and a flip aboutthe line F is indeed a rotation of measure π.

When F = H is a hyperplane, we can give an explicit for-mula for s(u) in terms of any nonnull vector w orthogonalto H .

We get

s(u) = u− 2(u · w)

‖w‖2 w.

Such reflections are represented by matrices called House-holder matrices , and they play an important role in nu-merical matrix analysis. Householder matrices are sym-metric and orthogonal.


Over an orthonormal basis (e1, . . . , en), a hyperplane re-flection about a hyperplane H orthogonal to a nonnullvector w is represented by the matrix

H = In − 2WW�

‖W‖2 = In − 2WW�

W�W,

where W is the column vector of the coordinates of w.

Since

pG(u) =(u · w)

‖w‖2 w,

the matrix representing pG is

WW�

W�W,

and since pH + pG = id, the matrix representing pH is

In − WW�

W�W.


The following fact is the key to the proof that every isom-etry can be decomposed as a product of reflections.

Lemma 7.1.3 Let E be any nontrivial Euclidean space.For any two vectors u, v ∈ E, if ‖u‖ = ‖v‖, then thereis an hyperplane H such that the reflection s about Hmaps u to v, and if u �= v, then this reflection isunique.

7.2. THE CARTAN–DIEUDONNE THEOREM FOR LINEAR ISOMETRIES 263

7.2 The Cartan–Dieudonne Theorem for Linear Isome-

tries

The fact that the group O(n) of linear isometries is gen-erated by the reflections is a special case of a theoremknown as the Cartan–Dieudonne theorem.

Elie Cartan proved a version of this theorem early in thetwentieth century. A proof can be found in his book onspinors [?], which appeared in 1937 (Chapter I, Section10, pages 10–12).

Cartan’s version applies to nondegenerate quadratic formsover R or C. The theorem was generalized to quadraticforms over arbitrary fields by Dieudonne [?].

One should also consult Emil Artin’s book [?], which con-tains an in-depth study of the orthogonal group and an-other proof of the Cartan–Dieudonne theorem.

First, let us recall the notions of eigenvalues and eigen-vectors.


Recall that given any linear map f :E → E, a vectoru ∈ E is called an eigenvector, or proper vector, orcharacteristic vector of f iff there is some λ ∈ K suchthat

f (u) = λu.

In this case, we say that u ∈ E is an eigenvector asso-ciated with λ.

A scalar λ ∈ K is called an eigenvalue, or proper value,or characteristic value of f iff there is some nonnullvector u �= 0 in E such that

f (u) = λu,

or equivalently if Ker (f − λid) �= {0}.

Given any scalar λ ∈ K, the set of all eigenvectors asso-ciated with λ is the subspace Ker (f − λid), also denotedas Eλ(f ) or E(λ, f ), called the eigenspace associatedwith λ, or proper subspace associated with λ.


Theorem 7.2.1 (Cartan-Dieudonne) Let E be a Eu-clidean space of dimension n ≥ 1. Every isometryf ∈ O(E) which is not the identity is the compositionof at most n reflections. For n ≥ 2, the identity is thecomposition of any reflection with itself.

Remarks .

(1) The proof of theorem 7.2.1 shows more than stated.

If 1 is an eigenvalue of f , for any eigenvector w associatedwith 1 (i.e., f (w) = w, w �= 0), then f is the compositionof k ≤ n− 1 reflections about hyperplanes Fi, such thatFi = Hi ⊕ L, where L is the line Rw, and the Hi aresubspaces of dimension n− 2 all orthogonal to L.

If 1 is not an eigenvalue of f , then f is the compositionof k ≤ n reflections about hyperplanes H,F1, . . . , Fk−1,such that Fi = Hi⊕ L, where L is a line intersecting H ,and theHi are subspaces of dimension n−2 all orthogonalto L.


u

h

w

λw

H

H1

Hi

Hk

LFi

Figure 7.2: An Isometry f as a composition of reflections, when 1 is an eigenvalue of f

w

f(w)

f(w) − wL⊥

H1

Hi

Hk−1

LFi

H

Figure 7.3: An Isometry f as a composition of reflections, when 1 is not an eigenvalue of f


(2) It is natural to ask what is the minimal number ofhyperplane reflections needed to obtain an isometry f .

This has to do with the dimension of the eigenspaceKer (f − id) associated with the eigenvalue 1.

We will prove later that every isometry is the compositionof k hyperplane reflections, where

k = n− dim(Ker (f − id)),

and that this number is minimal (where n = dim(E)).

When n = 2, a reflection is a reflection about a line, andtheorem 7.2.1 shows that every isometry in O(2) is eithera reflection about a line or a rotation, and that everyrotation is the product of two reflections about some lines.


In general, since det(s) = −1 for a reflection s, whenn ≥ 3 is odd, every rotation is the product of an evennumber ≤ n− 1 of reflections, and when n is even, everyimproper orthogonal transformation is the product of anodd number ≤ n− 1 of reflections.

In particular, for n = 3, every rotation is the product oftwo reflections about planes.

If E is a Euclidean space of finite dimension andf :E → E is an isometry, if λ is any eigenvalue of f andu is an eigenvector associated with λ, then

‖f (u)‖ = ‖λu‖ = |λ| ‖u‖ = ‖u‖ ,which implies |λ| = 1, since u �= 0.

Thus, the real eigenvalues of an isometry are either +1 or−1.


When n is odd, we can say more about improper isome-tries. This is because they admit −1 as an eigenvalue.When n is odd, an improper isometry is the composi-tion of a reflection about a hyperplane H with a rotationconsisting of reflections about hyperplanes F1, . . . , Fk−1

containing a line, L, orthogonal to H .

Lemma 7.2.2 Let E be a Euclidean space of finitedimension n, and let f :E → E be an isometry. Forany subspace F of E, if f (F ) = F , then f (F⊥) ⊆ F⊥

and E = F ⊕ F⊥.


Lemma 7.2.2 is the starting point of the proof that everyorthogonal matrix can be diagonalized over the field ofcomplex numbers.

Indeed, if λ is any eigenvalue of f , thenf (Eλ(f )) = Eλ(f ), and thus the orthogonal Eλ(f )⊥ isclosed under f , and

E = Eλ(f ) ⊕ Eλ(f )⊥.

The problem over R is that there may not be any realeigenvalues.


However, when n is odd, the following lemma shows thatevery rotation admits 1 as an eigenvalue (and similarly,when n is even, every improper orthogonal transforma-tion admits 1 as an eigenvalue).

Lemma 7.2.3 Let E be a Euclidean space.

(1) If E has odd dimension n = 2m + 1, then ev-ery rotation f admits 1 as an eigenvalue and theeigenspace F of all eigenvectors left invariant un-der f has an odd dimension 2p + 1. Furthermore,there is an orthonormal basis of E, in which f isrepresented by a matrix of the form(

R2(m−p) 00 I2p+1

)

where R2(m−p) is a rotation matrix that does nothave 1 as an eigenvalue.


(2) If E has even dimension n = 2m, then every im-proper orthogonal transformation f admits 1 as aneigenvalue and the eigenspace F of all eigenvectorsleft invariant under f has an odd dimension 2p+1.Furthermore, there is an orthonormal basis of E,in which f is represented by a matrix of the form(

S2(m−p)−1 00 I2p+1

)

where S2(m−p)−1 is an improper orthogonal matrixthat does not have 1 as an eigenvalue.

An example showing that lemma 7.2.3 fails for n even isthe following rotation matrix (when n = 2):

R =

(cos θ − sin θsin θ cos θ

)

The above matrix does not have real eigenvalues ifθ �= kπ.


It is easily shown that for n = 2, with respect to anychosen orthonormal basis (e1, e2), every rotation is rep-resented by a matrix of form

R =


)

where θ ∈ [0, 2π[, and that every improper orthogonaltransformation is represented by a matrix of the form

S =

(cos θ sin θsin θ − cos θ

)

In the first case, we call θ ∈ [0, 2π[ the measure of the an-gle of rotation of R w.r.t. the orthonormal basis (e1, e2).

In the second case, we have a reflection about a line, andit is easy to determine what this line is. It is also easyto see that S is the composition of a reflection about thex-axis with a rotation (of matrix R).


� We refrained from calling θ “the angle of rotation”,because there are some subtleties involved in defining

rigorously the notion of angle of two vectors (or two lines).

For example, note that with respect to the “opposite ba-sis” (e2, e1), the measure θ must be changed to 2π − θ(or −θ if we consider the quotient set R/2π of the realnumbers modulo 2π).

We will come back to this point after having defined thenotion of orientation (see Section 7.8).

It is easily shown that the group SO(2) of rotations inthe plane is abelian.


We can perform the following calculation, using some el-ementary trigonometry:

(cosϕ sinϕsinϕ − cosϕ

)(cosψ sinψsinψ − cosψ

)

=

(cos(ϕ + ψ) sin(ϕ + ψ)sin(ϕ + ψ) − cos(ϕ + ψ)

).

The above also shows that the inverse of a rotation matrix

R =


)

is obtained by changing θ to −θ (or 2π − θ).

Incidently, note that in writing a rotation r as the productof two reflections r = s2s1, the first reflection s1 can bechosen arbitrarily, since s2

1 = id, r = (rs1)s1, and rs1 isa reflection.


For n = 3, the only two choices for p are p = 1, whichcorresponds to the identity, or p = 0, in which case, f isa rotation leaving a line invariant.

u

R(u)

θ/2

D

Figure 7.4: 3D rotation as the composition of two reflections

This line is called the axis of rotation. The rotation Rbehaves like a two dimentional rotation around the axisof rotation.


The measure of the angle of rotation θ can be determinedthrough its cosine via the formula

cos θ = u ·R(u),

where u is any unit vector orthogonal to the direction ofthe axis of rotation.

However, this does not determine θ ∈ [0, 2π[ uniquely,since both θ and 2π − θ are possible candidates.

What is missing is an orientation of the plane (throughthe origin) orthogonal to the axis of rotation. We willcome back to this point in Section 7.8.

In the orthonormal basis of the lemma, a rotation is rep-resented by a matrix of the form

R =

cos θ − sin θ 0

sin θ cos θ 00 0 1

.


Remark : For an arbitrary rotation matrix A, since

a1 1 + a2 2 + a3 3

(the trace of A) is the sum of the eigenvalues of A, andsince these eigenvalues are cos θ + i sin θ, cos θ − i sin θ,and 1, for some θ ∈ [0, 2π[, we can compute cos θ from

1 + 2 cos θ = a1 1 + a2 2 + a3 3.

It is also possible to determine the axis of rotation (seethe problems).

An improper transformation is either a reflection about aplane, or the product of three reflections, or equivalentlythe product of a reflection about a plane with a rotation,and a closer look at theorem 7.2.1 shows that the axis ofrotation is orthogonal to the plane of the reflection.


Thus, an improper transformation is represented by amatrix of the form

S =

cos θ − sin θ 0

sin θ cos θ 00 0 −1

.

When n ≥ 3, the group of rotations SO(n) is not onlygenerated by hyperplane reflections, but also by flips (aboutsubspaces of dimension n− 2).

We will also see in Section 7.4 that every proper affinerigid motion can be expressed as the composition of atmost n flips, which is perhaps even more surprising!

The proof of these results uses the following key lemma.


Lemma 7.2.4 Given any Euclidean space E of di-mension n ≥ 3, for any two reflections h1 and h2

about some hyperplanes H1 and H2, there exist twoflips f1 and f2 such that h2 ◦ h1 = f2 ◦ f1.

Using lemma 7.2.4 and the Cartan-Dieudonne theorem,we obtain the following characterization of rotations whenn ≥ 3.

Theorem 7.2.5 Let E be a Euclidean space of di-mension n ≥ 3. Every rotation f ∈ SO(E) is thecomposition of an even number of flips f = f2k◦· · ·◦f1,where 2k ≤ n. Furthermore, if u �= 0 is invariant un-der f (i.e. u ∈ Ker (f − id)), we can pick the last flipf2k such that u ∈ F⊥

2k, where F2k is the subspace ofdimension n− 2 determining f2k.


Remarks :

(1) It is easy to prove that if f is a rotation in SO(3), ifD is its axis and θ is its angle of rotation, then f is thecomposition of two flips about linesD1 andD2 orthogonalto D and making an angle θ/2.

(2) It is natural to ask what is the minimal number offlips needed to obtain a rotation f (when n ≥ 3). Asfor arbitrary isometries, we will prove later that everyrotation is the composition of k flips, where

k = n− dim(Ker (f − id)),

and that this number is minimal (where n = dim(E)).

Hyperplane reflections can be used to obtain another proofof the QR-decomposition.


7.3 QR-Decomposition Using Householder Matrices

First, we state the result geometrically. When translatedin terms of Householder matrices, we obtain the fact ad-vertised earlier that every matrix (not necessarily invert-ible) has a QR-decomposition.

7.3. QR-DECOMPOSITION USING HOUSEHOLDER MATRICES 283

Lemma 7.3.1 Let E be a nontrivial Euclidean spaceof dimension n. Given any orthonormal basis(e1, . . . , en), for any n-tuple of vectors (v1, . . . , vn), thereis a sequence of n isometries h1, . . . , hn, such thathi is a hyperplane reflection or the identity, and if(r1, . . . , rn) are the vectors given by

rj = hn ◦ · · · ◦ h2 ◦ h1(vj),

then every rj is a linear combination of the vectors(e1, . . . , ej), (1 ≤ j ≤ n). Equivalently, the matrixR whose columns are the components of the rj overthe basis (e1, . . . , en) is an upper triangular matrix.Furthermore, the hi can be chosen so that the diagonalentries of R are nonnegative.

Remarks . (1) Since every hi is a hyperplane reflection orthe identity,

ρ = hn ◦ · · · ◦ h2 ◦ h1

is an isometry.


(2) If we allow negative diagonal entries in R, the lastisometry hn may be omitted.

(3) Instead of picking rk,k = ‖u′′k‖, which means that

wk = rk,k ek − u′′k,

where 1 ≤ k ≤ n, it might be preferable to pickrk,k = −‖u′′k‖ if this makes ‖wk‖2 larger, in which case

wk = rk,k ek + u′′k.

Indeed, since the definition of hk involves division by‖wk‖2, it is desirable to avoid division by very small num-bers.

Lemma 7.3.1 immediately yields the QR-decompositionin terms of Householder transformations.


Lemma 7.3.2 For every real n × n-matrix A, thereis a sequence H1, . . . , Hn of matrices, where each Hi

is either a Householder matrix or the identity, and anupper triangular matrix R, such that

R = Hn · · ·H2H1A.

As a corollary, there is a pair of matrices Q,R, whereQ is orthogonal and R is upper triangular, such thatA = QR (a QR-decomposition of A). Furthermore,R can be chosen so that its diagonal entries are non-negative.

Remarks . (1) Letting

Ak+1 = Hk · · ·H2H1A,

with A1 = A, 1 ≤ k ≤ n, the proof of lemma 7.3.1 can beinterpreted in terms of the computation of the sequenceof matrices A1, . . . , An+1 = R.


The matrix Ak+1 has the shape

Ak+1 =

× × × uk+11 × × × ×

0 × ... ... ... ... ... ...0 0 × uk+1

k × × × ×0 0 0 uk+1

k+1 × × × ×0 0 0 uk+1

k+2 × × × ×... ... ... ... ... ... ... ...0 0 0 uk+1

n−1 × × × ×0 0 0 uk+1

n × × × ×

where the (k + 1)th column of the matrix is the vector

uk+1 = hk ◦ · · · ◦ h2 ◦ h1(vk+1),

and thusu′k+1 = (uk+1

1 , . . . , uk+1k ),

andu′′k+1 = (uk+1

k+1, uk+1k+2, . . . , u

k+1n ).

If the last n− k − 1 entries in column k + 1 are all zero,there is nothing to do and we let Hk+1 = I .


Otherwise, we kill these n− k− 1 entries by multiplyingAk+1 on the left by the Householder matrix Hk+1 sending

(0, . . . , 0, uk+1k+1, . . . , u

k+1n ) to (0, . . . , 0, rk+1,k+1, 0, . . . , 0),

whererk+1,k+1 =

∥∥(uk+1k+1, . . . , u

k+1n )

∥∥ .(2) If we allow negative diagonal entries in R, the matrixHn may be omitted (Hn = I).

(3) If A is invertible and the diagonal entries of R arepositive, it can be shown that Q and R are unique.


(4) The method allows the computation of the determi-nant of A. We have

det(A) = (−1)mr1,1 · · · rn,n,where m is the number of Householder matrices (not theidentity) among the Hi.

(5) The condition number of the matrix A is preserved(see Strang [?]). This is very good for numerical stability.

We conclude our discussion of isometries with a brief dis-cussion of affine isometries.

chapter 7 the cartan–dieudonn´e theoremcis610/cis61009sl5.pdf256 chapter 7. the...

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